In general, a ruled surface can be thought of as set of points created by moving
a straight line. One way of creating the equation of a ruled surface is to consider
two curves where both curves are defined in terms of a parameter tand represented
by the position vectors r 1 (t)and r 2 (t)as illustrated in the figure 7-12.
Figure 7-12. Generating a ruled surface using two curves.
For a fixed value of the parameter t, one can draw a straight line between the two
points r 1 (t)and r 2 (t)as illustrated in the figure 7-12. If r is the position vector to a
general point on this line it can be represented by the equations
r =r (t, u ) = (1 −u)r 1 (t) + ur 2 (t) −u 0 ≤u≤u 0 (7 .50)
where uis a parameter and u 0 is some specified constant. Note that when u= 0 ,
then r =r 1 and when u= 1,r =r 2. As the parameter tchanges the line sweeps out
a surface.
Ruled surfaces can be observed on cylinders, cones, hyperboloids of one sheet, as
well as elliptic and hyperbolic paraboloids. Ruled surfaces have been studied since
the time of the early Greeks and many architectural structures can be described as
ruled surfaces.
